This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Evaluate the fourth limit. _x 9 sqrt(x)-3x-9 Substituting x=9 directly gives sqrt(9)-39-9 = (3-3)/(0) = (0)/(0), which is an indeterminate form. To resolve this, we multiply the numerator and the denominator by the conjugate of the numerator, which is (sqrt(x)+3). _x 9 sqrt(x)-3x-9 × sqrt(x)+3sqrt(x)+3 _x 9 (sqrt(x))^2 - 3^2(x-9)(sqrt(x)+3) _x 9 (x - 9)/((x-9)(sqrt(x)+3)) Since x 9, x ≠ 9, so we can cancel the common factor (x-9): _x 9 (1)/(sqrt(x)+3) Now, substitute x=9: (1)/(sqrt(9)+3) = (1)/(3+3) = (1)/(6) The value of the fourth limit is (1)/(6). Step 2: Evaluate the fifth limit. _t 0 sqrt(t^2+9)-3t^2 Substituting t=0 directly gives sqrt(0^2+9)-30^2 = sqrt(9)-30 = (3-3)/(0) = (0)/(0), which is an indeterminate form. To resolve this, we multiply the numerator and the denominator by the conjugate of the numerator, which is (sqrt(t^2+9)+3). _t 0 sqrt(t^2+9)-3t^2 × sqrt(t^2+9)+3sqrt(t^2+9)+3 _t 0 (sqrt(t^2+9))^2 - 3^2t^2(sqrt(t^2+9)+3) _t 0 (t^2+9 - 9)/(t^2(sqrt(t^2+9)+3)) _t 0 (t^2)/(t^2(sqrt(t^2+9)+3)) Since t 0, t ≠ 0, so we can cancel the common factor t^2: _t 0 (1)/(sqrt(t^2+9)+3) Now, substitute t=0: (1)/(sqrt(0^2+9)+3) = (1)/(sqrt(9)+3) = (1)/(3+3) = (1)/(6) The value of the fifth limit is (1)/(6). That's 2 down. 3 left today — send the next one.